Lab 2 Geometrical Optics March 22, 202 This material will span much of 2 lab periods. Get through section 5.4 and time permitting, 5.5 in the first lab. Basic Equations Lensmaker s Equation for a thin lens of index n 2 immersed in a medium of index n, with radii of curvature of R and R 2 f = n ( 2 n ) () n R R 2 Lens equation (or mirror) focal length, f, object distance, s o, and image distance s i s o + s i = f (2) Converging lenses have positive focal lengths and diverging lenses have negative focal lengths. Real objects and images have positive distances, while virtual objects and images have negative distances. Lateral magnification is defined in terms of the heights of the object and the image, and is negative for an inverted image. It is m = h i h o = s i s o (3) Power of a lens is the inverse of the focal length, and is measured in diopters, which are inverse meters. If two thin lenses are in close proximity, their powers add, P combined = P + P 2 f combined = f + f 2 (4)
Near point and far point and depth of field for object at s o, lens of focal length f and stop diameter D, hence aperture A = f/d s N = s o f 2 f 2 + Ad(s o f) s F = s o f 2 f 2 Ad(s o f) (5) DOF = s F s N = 2s of 2 Ad(s o f) f 4 A 2 d 2 (s o f) 2 (6) 2 Handling Optical Components Avoid fingerprints! The optical components are stored with lens paper around them. Handle only by the edges, and minimally. The components we have are inch diameter and are held in lollypop mounts a lens holder mounted on an optical post. Components are secured by an Allen screw. When done, store the components back in their cases. 3 The track The optical posts fit into an post holder mounted on a platform that moves along a track. The platform is.875 inches long, and the post is centered on the platform. The location of the platform can be measured using the scale on the track measure as precisely as possible. You will need to account for the offset between the platform edge and center of the post. Record the raw data that you see, and do the offset in your lab book: for example if the offset is 2.34 cm and the edge of the platform is at 5.68 cm, then the center is at (5.68-2.34 = 3.34 cm. It is quite easy to make a mistake, and if you add/subtract in your head and then record you will have no way of fixing the result. 4 Uncertainties Locations of object and lens are easy and precise (a fraction of mm), but the location of the image has issues. You must decide when the image is sharpest. Record image uncertainties for all images. I will not ask you to do any propagation of error, but you can use the measured uncertainty in image position when comparing with the predicted. 2
5 Experimental 5. Finding the focal length of a lens (a) Use the nominal +00 mm lens. Mount the lens and the ground-glass diffusing screen on your track. Go into the hall and adjust the screen until you get a sharp image of the window at the far end of the hall. Assuming the window to be infinitely far away, s o =, the image distance equals the focal length, s i = f. Hmm. Infinitely far away? Determine the distance to the window to within 50 cm (hint, count tiles) and determine f by using Equation 2. What is the error introduced by using s =? (b) Now put a diffuser over the light source, and use the crossed arrow slide as an object. For object distances of 200, 220, 250, 280,300, 350, and 400 mm, find the image distance when the image is judged sharpest. Use Equation 2 to find the focal length in each case. Also measure the height of the object and image in each case and determine the magnification from this data (negative for inverted object.) Compare the magnification from this direct measure to the magnification from Equation 3. Characterize the image (erect/inverted, real/virtual, magnified/reduced.) Compare two methods of finding the focal length. Discuss which measurement you trust the most. Record the focal length that you have settled on for the nominal 00 mm lens. 5.2 Focal length from Lensmakers formula, Equation. Use a plano-convex +50 mm lens. It has index of refraction of.55. Find the focal length experimentally. Use calipers to measure the diameter of the lens, the thickness at the center, and at the thickness at the edge of the lens. From these measurements you should be able to find the radius of curvature of the convex side of the lens. Show your derivation in your lab book. Calculate the focal length from the Lensmaker s equation and compare with your measured value. 3
5.3 Thin lenses close together Two thin lenses that are close together should have a combined power equal to the sum of the powers of the lenses, Equation 4. (a) Mount your plano-convex +50 mm lens in the same holder as your +00 mm lens. Measure the focal length of the combination. Compare your result with that calculated from Equation 4 (b) Mount the +50 mm and -00 mm lenses in the same mount and measure the effective focal length of the combination. Use Equation 4 to find the focal length of the negative lens. 5.4 Virtual Images Real images can be located fairly easily since they actual appear on a screen. Virtual images are trickier to locate experimentally. Use a stick as an object and locate it about 30 cm from a plane mirror. Put a second stick on a platform behind the mirror and locate the virtual image by using parallax. I will demonstrate. Compare your measured image distance to the expected value. 5.5 First two-lens system I may change some numbers by Monday Be sure you have measured the focal lengths of the +50 mm and +00 mm lenses. Put the source 5.0 cm in front of the +00 mm lens. Use the screen to locate the image and record the location of the screen and use it to find the image distance for the first lens. Remove the screen and place the +50 mm lens 7.0 cm to the right of the image location you just found. Find the final image location and the overall magnification of the system. Analyze the system and see if it is consistent with your measurements. 5.6 Second two-lens system I may change some numbers by Monday Start with the 00 mm lens located 5.0 cm to the right of the object. Locate the real image S. Now place the -50 mm lens 3.0 cm to the left of the image you just found. Locate the final image and determine its magnification. 4
Analyze the system and determine the focal length of the negative lens. Compare magnification to its calculated value. 5.7 Depth of Field Use the light source with crossed arrow target, nominal 00 mm lens, and a ground glass screen. Suppose a lens of focal length f with a stop of diameter D has an object located at object distance s. There will be a sharp image at some location. If the f-stop (aperature) of the lens is A = f/d, then there is a range of distances that will also be in acceptible focus. In class we found expressions for the near and far object locations, s N, s F, and depth of field, DOF, for a given circle of confusion diameter d. Start with an unstopped lens and determine its aperture (f-number). Place the light source with diffuser and crossed arrow target near one end of a track, with the carrier at about 30 cm. Use a nominal 00 mm lens (you measured its actual focal length earlier.) Place the screen carrier at about the 20 cm mark. Adjust the lens position to get a sharp image. Arrange the telescope so you can see the image and adjust the lens position to get the sharpest image. Lock the lens and screen into position. Now move the object closer to the lens until the image is just out of focus (a judgement call). This lets you get s N. Move the object further away from the lens until the just out of focus and find s F. Each person should make these measurements twice so you can get a good average. Calculate the depth of field from these. Now reduce the aperture by using a hole from a 3-hole punch. Measure the diameter and calculate the aperture. Put the aperture in front of the lens and make measurents so you can find s N and s F and calculate depth of field. Finally use the 2 mm aperture in the kit and repeat the measurements. (I will let you know if this is possible in lab.) Calculate the diameter of the circle of confusion, d, using s N and s F Discuss the consistency of your results. for each aperture. 5.8 Chromatic Aberration In the first lab you measured the dispersion of glass in a prism. Since the index depends on wavelength, the focal length should be different for different wavelengths. This produces chromatic aberration in images. 5
We have 3 filters on hand, a (650 ± 20) nm red, a (550 ± 20) nm green, and a (450 ± 20) nm blue/violet. Place the light source with target at one end of the track (near 0 cm). Place the red filter just in front of the object, and your 00 mm lens (nominal) 30 cm from the object. Adjust the screen to get the sharpest image, and record the image distance. Each member should repeat this measurement twice, starting with the screen at an out-of-focus point. Repeat for the green and blue-violet filters. For each wavelength, determine the focal length. Does the index of the glass increase or decrease as you go from red to blue-violet? Explain your conclusion. Is this consistent with the dispersion lab? 6